A difference method for solving the nonlinear $q$-factional differential equations on time scale

TL;DR

This paper introduces a new difference method for solving nonlinear q-fractional differential equations on time scales, providing stability, convergence, and high accuracy validated by numerical experiments.

Contribution

The paper proposes a novel difference formula for discretizing q-fractional derivatives on time scales, with proven stability, convergence, and error bounds, advancing numerical solutions for such equations.

Findings

The difference formula is unconditionally stable.

The method achieves high accuracy in numerical experiments.

The approach ensures unique existence and stability of solutions.

Abstract

The $q$-fractional differential equation usually describe the physics process imposed on the time scale set $T_q$. In this paper, we first propose a difference formula for discretizing the fractional $q$-derivative $^cD_q^\alpha x(t)$ on the time scale set $T_q$ with order $0<\alpha<1$ and scale index $0<q<1$. We establish a rigours truncation error boundness and prove that this difference formula is unconditionally stable. Then, we consider the difference method for solving the initial problem of $q$-fractional differential equation: $^cD_q^\alpha x(t)=f(t,x(t))$ on the time scale set. We prove the unique existence and stability of the difference solution and give the convergence analysis. Numerical experiments show the effectiveness and high accuracy of the proposed difference method.